Question: Simplify the following expression: $a = \dfrac{24q + 48}{-40q + 72}$ You can assume $q \neq 0$.
Answer: Find the greatest common factor of the numerator and denominator. The numerator can be factored: $24q + 48 = (2\cdot2\cdot2\cdot3 \cdot q) + (2\cdot2\cdot2\cdot2\cdot3)$ The denominator can be factored: $-40q + 72 = - (2\cdot2\cdot2\cdot5 \cdot q) + (2\cdot2\cdot2\cdot3\cdot3)$ The greatest common factor of all the terms is $8$ Factoring out $8$ gives us: $a = \dfrac{(8)(3q + 6)}{(8)(-5q + 9)}$ Dividing both the numerator and denominator by $8$ gives: $a = \dfrac{3q + 6}{-5q + 9}$